I find Jungle Eve to be much more effective in ranked than Mid lane Eve. As jungler she has the roam potential she was basically designed for. The best way I've found to play Eve is this (Remade Eve with the S3 item and jungle remake, TL;DR version at the bottom):
I use my standard "AP Jungle" masteries and might be unusual I admit at 16-14-0.
Offensive I take the increased monster damage in Butcher and the 2% CDR in Sorcery. The rest of the 12 points go towards all AP masteries on the right side and Havoc
For Defensive I take the improves smite in Summoner's resolve. 4 points in durability and 2 in perserverance. 3 points total for Tough Skin and Bladed armor. Then the last 4 points all in Hardiness And Veteran's Scars for better sustainability in the starting jungle
The runes I use are NOT optimal Jungle runes but this is what I have for reference.
Scaling AP Marks
Flat Armor Seals
Flat AP Glyphs
Flat AP Quints
Totals: 25.6 Flat AP, 16.82 AP @ lvl 18, 12.7 Flat Armor.
This is my typical Jungle route for Eve specifically:
-Start hunter's machete + 5 pots. Get Q first
-Go to wolves and pound Q and AA the big wolf down first.
-Have mid-lane damage blue golem to start the leash. I like to let top lane tank it for a few hits while I stand back hitting it with Q.
-Once top lane lane leaves Smite blue and take out the small lizards. You should level up and take E here. Might want to take a pot as well now on the way to Wraiths.
-Make sure you go invis on your way to Wraiths so you get some mana back also.
-Q as you walk up to the wraiths and E the big one. Continue with Q and AA until dead.
-Same with smaller golems. Q to start and E when you get close. Continue Q and AA's. Use a pot or two here also before taking on Red.
-If you level here you can either take second point in Q if you want to continue jungling or W if you want to gank (which is what I recommend doing).
-Go invis for a bit before taking on Red buff to get some mana and have smite come off CD.
-Open with E on the Lizard Elder and pound Q again until dead.
-Gank a lane if someone is pushed (and you have quite a bit of health left ~350 at least or another pot left) or go back to base.
(Just a note: I prefer repeated ganks afer I go to base over continually farming the jungle. Though if the enemy is not pushed far or your team is winning their lane for the most part, Jungling may be your only option. Also, after the first blue, you don't have as much of a need for it as long as you know how to keep up your mana with your passive. Give blue to Mid whenever possible)
Another effective but situational/optional route is:
-start at opponent's red buff (assuming the enemy jungler will for sure start at their blue).
-Take red with help of your mid or top lane. You should level up here and take E.
-Wait in tri-bush by top lane and gank ASAP for an almost guaranteed kill against most top laners
-Go back to base and continue with standard route above.
(Use mainly if you fear top lane will be at a disadvantage due to counter-picking or they face the rare duo top lane)
Good Start Items:
Hunter's machete > spirit stone > sell later when you need an inventory space
Boots > Boots of mobility
Items I try to get every game:
Haunting Guise > Liandry's Torment
Sheen > Lich Bane
Hextech Revolver > Hextech Gunblade
Items for more defenses
Negatron Cloak > Abyssal Scepter
Chainmail > Zhonya's Hourglass
Giant's Belt > Rylai's Crystal Scepter
Catalyst The Protector > Rod of Ages
Items for the fed Eve
Needlessly Large Rod > Rabbadon's Deathcap or DeathFire Grasp (Still useful after nerfs but no longer A must buy.)
Blasting Wand > Void Staff
Other Items that may do well but I have not tried myself yet
Tear of the Goddess > Archangel's Staff > Seraph's Embrace (With max bonus Mana)
Stinger > Nashor's Tooth
Will of the Ancients (instead of Hextech Gunblade)
Blasting Wand > Guinsoo's Rageblade
Stay Invisible as you approach the enemy. You don't need to stay in the bush like other jungler's because of the invisibility. Open up with E and Repeated Q's. Don't use W unless they start to get away from you or flash away. I don't use my ult unless I need the extra damage. Best used when you first engage for the slow.
To start off a teamfight I like to hit as many people as posible with my ulti. After, you can either use your shield to absorb damage while you E > Q their carry to death, or back off for a bit and wait till the enemy uses more of their abilities.
If you stay, try to E their Highest damage dealer as often as possible. Also Mash Q trying to hit as many as possible
If you get caught you can sprint out with W pretty quickly and get out of slows as well.
If you back off after using your ult, I try to get between the enemy and their base to pick off anyone who gets away with low health.
That's the way I like to play her and am generally incredibly successful with it.
Runes: Flat AP quint and blue. Flat armor yellow. Scaling AP red.
Masteries: 16-14-0 with AP boosts in offensive. Minion damage reduction in defense along with armor and health / health regen.
Route: wolves, blue, wraiths, small golems, red, gank or go base
Items: Start Hunter's + 5 pots, upgrade to Spirit Stone (sell later), Boots of Mobility Lich Bane, Liandry's Torment, Hextech Gunblade, Deathfire Grasp, Rabbadon's Deathcap.
Your combo is E > Q > Q > Q > Q > enemy is dead. Throw in W if they try to escape your wrath. If they are too stubborn to die or too fast for your team (not you, your too fast for anyone. Even Teemo), show em your ult and proceed to kill.
I think that's everything you need to know and be successful with the Jungle-is-Superior-to-Laning Evelynn.